Finite Group Databasebeta

← #28344140

Subgroups — #28344140

Order 13696.

448 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20292569C212yesmaximal, complemented
2#20292554C212yesmaximal, complemented
3#20292544C211yesmaximal, characteristic
4#20292519C212yesmaximal, complemented
5#20292505C212yesmaximal
6#20292501C212yesmaximal, complemented
7#20292498C212yesmaximal, complemented
8#20292492C211yesmaximal, characteristic, complemented
9#400×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C4×D107C2212yes—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×C4⋊Dic107C2212yes—
14C2×Dic107⋊C4C2212yes—
15C2×(C2×D107)⋊C4C2212yes—
16C2×(C2×D107)⋊C4C2212yes—
17C2×(C2×D107)⋊C4C2212yes—
18C2×(C2×D107)⋊C4C2212yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×C4×Dic107C2211yescharacteristic
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2211yescharacteristic
26C2×C4⋊C4×C107C2211yescharacteristic
27C2×C4⋊C4×C107C2211yescharacteristic
28#95×C107C2212yes—
29#95×C107C2212yes—
30#95×C107C2211yescharacteristic
31C23×D107C2312yes—
32C2×D428—24no—
33C2×D428—41no—
34C22×Dic107C2312yes—
35C22×Dic107C2312yes—
36C22×Dic107C2312yes—
37C2×C4×D107—42no—
38C2×C4×D107—42no—
39C2×C4×D107—42no—
40C4⋊Dic107—42no—
41C4⋊Dic107—41no—
42Dic107⋊C4—42no—
43(C2×D107)⋊C4—24no—
44(C2×D107)⋊C4—24no—
45(C2×D107)⋊C4—24no—
46(C2×D107)⋊C4—42no—
47(C2×D107)⋊C4—42no—
48C4×Dic107—22no—
49C22×C4×C107C2312yes—
50C22×C4×C107C2311yescharacteristic
51C22×C4×C107C2312yes—
52C22×C4×C107C2311yescharacteristic
53C22×C4×C107C2311yescharacteristic
54C4⋊C4×C107—41no—
55C4⋊C4×C107—41no—
56C23×C107C2411yescharacteristic
57C22×D107—22no—
58C22×D107—22no—
59C22×D107—22no—
60C22×D107—22no—
61C22×D107—22no—
62C22×D107—22no—
63C22×D107#4114yes—
64C2×C4×C107—21no—
65C2×C4×C107—21no—
66C2×C4×C107—22no—
67C2×C4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107—21no—
70C2×C4×C107—22no—
71C2×C4×C107—21no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107#4112yes—
77C2×Dic107—22no—
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86D428—44no—
87C4×D107—48no—
88C22×C107C2×#4112yes—
89C22×C107C2×#4111yescharacteristic
90C22×C107H(5,𝔽2)11yescharacteristic
91C22×C107H(5,𝔽2)12yes—
92C22×C1072−1+411yescharacteristic
93C2×D107—24no—
94C2×D107—24no—
95C2×D107—24no—
96C2×D107—42no—
97C2×D107—42no—
98C2×D107—42no—
99C2×D107—42no—
100C4×C107—22no—
101C4×C107—42no—
102C4×C107—41no—
103C4×C107—41no—
104C4×C107—42no—
105Dic107—42no—
106Dic107—42no—
107Dic107—42no—
108C2×C107#53912yes—
109C2×C107#53911yescharacteristic
110C2×C107#55111yescharacteristic
111C2×C107#55312yes—
112C2×C107H(3,𝔽4)11yescharacteristic
113D107—44no—
114#2782—1071nomaximal, Sylow
115C107#278211yescharacteristic, complemented, Sylow
116#400—1072no—
117#400D10711yescharacteristic, complemented
118#397—1072no—
119#394—1071no—
120#388—1072no—
121#386—1072no—
122#385—1072no—
123#385—1072no—
124#382—1071no—
125C22×D4—1071no—
126C23×C4—1072no—
127C2×C4⋊C4—1072no—
128C2×C4⋊C4—1071no—
129C2×C4⋊C4—1072no—
130C2×C4⋊C4C2×D10711yescharacteristic
131C2×C4⋊C4C2×D10711yescharacteristic
132C2×C22⋊C4—1072no—
133C2×C22⋊C4—1072no—
134C2×C22⋊C4—1072no—
135C2×C22⋊C4—1072no—
136C2×C22⋊C4—1072no—
137C2×C42—1071no—
138#95—1072no—
139#95—1072no—
140#95—1072no—
141#95—1072no—
142#95—1071no—
143#95C2×D10712yes—
144#95C2×D10712yes—
145#95C2×D10711yescharacteristic
146C24—1072no—
147C2×D4—4281no—
148C2×D4—2144no—
149C22×C4—1072no—
150C22×C4—1072no—
151C22×C4—1072no—
152C22×C4—4282no—
153C22×C4—4282no—
154C22×C4—4282no—
155C22×C4C22×D10711yescharacteristic
156C22×C4C22×D10711yescharacteristic
157C22×C4C22×D10712yes—
158C22×C4C22×D10712yes—
159C22×C4C22×D10711yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—4282no—
163C4⋊C4—4281no—
164C4⋊C4—4282no—
165C22⋊C4—4282no—
166C22⋊C4—4282no—
167C22⋊C4—2144no—
168C22⋊C4—2144no—
169C22⋊C4—2144no—
170C42—2142no—
171C23—1074no—
172C23—2142no—
173C23—2142no—
174C23—2142no—
175C23—2142no—
176C23—2142no—
177C23—2142no—
178C23C23×D10711yescharacteristic
179D4—4284no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—4288no—
193C2×C4—2142no—
194C2×C4—2142no—
195C2×C4—2142no—
196C2×C4—2142no—
197C2×C4—2142no—
198C2×C4—2142no—
199C2×C4—2142no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4#1313381912yes—
203C22—4282no—
204C22—4282no—
205C22—4282no—
206C22—4282no—
207C22—2144no—
208C22—2144no—
209C22—2144no—
210C22#41×D10712yes—
211C22C2×#1313381911yescharacteristic
212C22#1681215411yescharacteristic
213C22#1681215512yes—
214C22#1681215611yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—42no—
220C4—4282no—
221C4—4282no—
222C4—4282no—
223C2—4284no—
224C2#2029321812yes—
225C2#2029322611yescharacteristic
226C2#2029330511yescharacteristic
227C2#2029333012yes—
228C2#2029336911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#400×C107C2
prime radicalC22×D428C22
hypercenter#400D107
hypocenterC107#2782
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #400

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20292569C2semidirect
2. #20292554C2semidirect
4. #20292519C2semidirect
6. #20292501C2semidirect
7. #20292498C2semidirect
8. #20292492C2semidirect
9. #400×C107C2semidirect
115. C107#2782semidirect
117. #400D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202925692
#202925542
#202925442
#202925192
#202925052
#202925012
#202924982
#202924922
#400×C1072
#2782107